The boundary-Wecken classification of surfaces

Let $X$ be a compact $2$ -manifold with nonempty boundary $\partial X$ and let $f : (X, \partial X) \to (X, \partial X)$ be a boundary-preserving map. Denote by $M F_{\partial} [f]$ the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to $f$ . The relative Nielsen number $N_{\partial} (f)$ is the sum of the number of essential fixed point classes of the restriction $\bar{f} : \partial X \to \partial X$ and the number of essential fixed point classes of $f$ that do not contain essential fixed point classes of $\bar{f}$ . We prove that if $X$ is the Möbius band with one (open) disc removed, then $M F_{\partial} [f] - N_{\partial} (f) \leq 1$ for all maps $f : (X, \partial X) \to (X, \partial X)$ . This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If $X$ is the disc, annulus or Möbius band, then $X$ is boundary-Wecken, that is, $M F_{\partial} [f] = N_{\partial} (f)$ for all boundary-preserving maps. If $X$ is the disc with two discs removed or the Möbius band with one disc removed, then $X$ is not boundary-Wecken, but $M F_{\partial} [f] - N_{\partial} (f) \leq 1$ . All other surfaces are totally non-boundary-Wecken, that is, given an integer $k \geq 1$ , there is a map $f_{k} : (X, \partial X) \to (X, \partial X)$ such that $M F_{\partial} [f_{k}] - N_{\partial} (f_{k}) \geq k$ .

Keywords

boundary-Wecken, relative Nielsen number, punctured Möbius band, boundary-preserving map

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 54H25, 57N05

References

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Publication

Received: 21 November 2002

Revised: 15 October 2003

Accepted: 26 November 2003

Published: 7 February 2004

Authors

Robert F Brown
Department of Mathematics University of California Los Angeles CA 90095-1555 USA

Michael R Kelly
Department of Mathematics and Computer Science Loyola University 6363 St Charles Avenue New Orleans LA 70118 USA

Volume 4, issue 1 (2004)

Robert F Brown and Michael R Kelly

Abstract